arXiv 1405.35282014-05-14nlin.CDcond-mat.stat-mech

长程相互作用的Fermi-Pasta-Ulam模型：动力学和热统计学

Fermi-Pasta-Ulam model with long-range interactions: Dynamics and thermostatistics

Helen Christodoulidi, Constantino Tsallis, Tassos Bountis

备注：: 8 pages, 5 fugures
DOI：: 10.1209/0295-5075/108/40006

摘要：

本文介绍并数值研究了一维Fermi-Pasta-Ulam（FPU）$β-$模型的长程相互作用推广。标准的四次相互作用通过一个衰减为$1/r^α$（$α\ge 0$）的耦合常数来推广（其强度由$b>0$表征）。在$α\to\infty$极限下，我们恢复了原始FPU模型。通过经典的分子动力学计算，我们证明了：（i）对于$α\geq 1$，随着振子个数的增加，最大李雅普诺夫指数保持为正（因此产生遍历性），而对于$0 \le α<1$，它渐近地减少为$N^{- κ(α)}$（违反遍历性）;（ii）对于足够大的$α$，时间平均速度的分布是麦克斯韦分布，而$q$ -高斯分布则很好地接近它，当$α$从0增加到接近1时，指数$q(α)$从大约1.5单调地减少到1（高斯）。对于足够小的$α$，整个图像与时间$t_c$处从$q$ -统计到玻尔兹曼-吉布斯（BG）热统计的交叉一致。更准确地说，我们构建了一个“相图”的系统，在该系统中，这种交叉发生通过一个前沿的形式$1/N \propto b^δ/t_c^γ$与$γ>0$和$δ>0$，在这样一种方式，$q=1$（$q>1$）的行为占主导地位的$\lim_{N \to\infty} \lim_{t \to\infty}$订购（$\lim_{t \to\infty} \lim_{N \to\infty}$订购）。

英文摘要：

We introduce and numerically study a long-range-interaction generalization of the one-dimensional Fermi-Pasta-Ulam (FPU) $β-$ model. The standard quartic interaction is generalized through a coupling constant that decays as $1/r^α$ ($α\ge 0$)(with strength characterized by $b>0$). In the $α\to\infty$ limit we recover the original FPU model. Through classical molecular dynamics computations we show that (i) For $α\geq 1$ the maximal Lyapunov exponent remains finite and positive for increasing number of oscillators $N$ (thus yielding ergodicity), whereas, for $0 \le α<1$, it asymptotically decreases as $N^{- κ(α)}$ (consistent with violation of ergodicity); (ii) The distribution of time-averaged velocities is Maxwellian for $α$ large enough, whereas it is well approached by a $q$-Gaussian, with the index $q(α)$ monotonically decreasing from about 1.5 to 1 (Gaussian) when $α$ increases from zero to close to one. For $α$ small enough, the whole picture is consistent with a crossover at time $t_c$ from $q$-statistics to Boltzmann-Gibbs (BG) thermostatistics. More precisely, we construct a "phase diagram" for the system in which this crossover occurs through a frontier of the form $1/N \propto b^δ/t_c^γ$ with $γ>0$ and $δ>0$, in such a way that the $q=1$ ($q>1$) behavior dominates in the $\lim_{N \to\infty} \lim_{t \to\infty}$ ordering ($\lim_{t \to\infty} \lim_{N \to\infty}$ ordering).

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文